2. An operation is defined between the elements of P and V, which is called scalar multiplication (also called quantity multiplication), that is, any element α in V and any element K in P correspond to the unique element kα in V according to certain rules, which is called the product of K and α.
3. Addition and scalar multiplication satisfy the following conditions:
1) α+β=β+α, for any α, β ∈ v.
2) α+(β+γ)=(α+β)+γ, for any α, β, γ∈V 。
3) There is an element 0∈V, which has α+0=α for all α∈V, and element 0 is called the zero element of V. 。
4) For any α∈V, there is a negative element β∈V that makes α+β=0, and β is called α and recorded as-α.
5) For the unit element 1 in P, there is 1α=α(α∈V).
6) For any k, l∈P, α ∈ v, there exists (kl)α=k(lα).
7) For any k, l∈P, α∈V, there is (k+l)α=kα+lα.
8) For any k∈P, α, β∈V, there is k(α+β)=kα+kβ,
Then v is called a linear space or a vector space over the field p.
Extended data:
If both V and W are vector spaces over field F, a linear transformation or "linear mapping" from V to W can be set. These mappings from v to w all have one thing in common, that is, to keep sum and quotient. This set contains all the linear mappings from V to W, which are described by L(V, W), and it is also a vector space over field F. When V and W are determined, the linear mappings can be represented by a matrix.
Isomorphism is a one-to-one linear mapping. If v and w are isomorphic, we call these two spaces isomorphic; Every n-dimensional vector space on field f is isomorphic to vector space F.
Learning vector space will naturally involve some extra structures. The additional structure is as follows:
1, the concept of real or complex vector space plus length. Norms are called normed vector spaces.
2. The concept of real or complex vector space plus length and angle is called inner product space.
3. A vector space plus topological coincidence operation (addition and scalar multiplication are continuous mappings) is called topological vector space.
4. Vector space plus bilinear operator (defined as vector multiplication) is a field algebra.
References:
Baidu encyclopedia-vector space